Judging the order of numbers relies on familiarity rather than activating the mental number line

The presence of the reverse distance effect depends on the familiarity of the sequences being processed

Number order processing is thought to be characterised by a reverse distance effect whereby consecutive sequences (e.g., 1-2-3) are processed faster than non-consecutive sequences (e.g., 1-3-5). However, there is accumulating evidence that the reverse distance effect is not consistently observed. In this context, the present study investigated whether the presence of the reverse distance effect depends on the familiarity of the sequences being processed. Supporting this proposal, Experiment 1 found that the reverse distance effect was only present when the presented consecutive sequences were considerably more familiar than the presented non-consecutive sequences. Additionally, the sequence 1-2-3 has been suggested to play a pivotal role in the presence of the reverse distance effect due to being both the most familiar and fastest processed sequence. However, it is contested whether 1-2-3 is processed fast because it is familiar or simply because it can typically be verified as ordered from only its first two digits. Supporting the familiarity explanation, Experiments 2 and 3 found that 1-2-3 was processed characteristically fast regardless of whether it could be verified from its first two digits. Taken together, these findings suggest that sequence familiarity plays a critical role in the presence or absence of the reverse distance effect.

Concepts of order: Why is ordinality processed slower and less accurately for non-consecutive sequences?

Both adults and children are slower at judging the ordinality of non-consecutive sequences (e.g., 1-3-5) than consecutive sequences (e.g., 1-2-3). It has been suggested that the processing of non-consecutive sequences is slower because it conflicts with the intuition that only count-list sequences are correctly ordered. An alternative explanation, however, may be that people simply find it difficult to switch between consecutive and non-consecutive concepts of order during order judgement tasks. Therefore, in adult participants, we tested whether presenting consecutive and non-consecutive sequences separately would eliminate this switching demand and thus improve performance. In contrast with this prediction, however, we observed similar patterns of response times independent of whether sequences were presented separately or together (Experiment 1). Furthermore, this pattern of results remained even when we doubled the number of trials and made participants explicitly aware when consecutive and non-consecutive sequences were presented separately (Experiment 2). Overall, these results suggest slower response times for non-consecutive sequences do not result from a cognitive demand of switching between consecutive and non-consecutive concepts of order, at least not in adults.

Familiar Sequences Are Processed Faster Than Unfamiliar Sequences, Even When They Do Not Match the Count-List

In order processing, consecutive sequences (e.g., 1-2-3) are generally processed faster than nonconsecutive sequences (e.g., 1-3-5) (also referred to as the reverse distance effect). A common explanation for this effect is that order processing operates via a memory-based associative mechanism whereby consecutive sequences are processed faster because they are more familiar and thus more easily retrieved from memory. Conflicting with this proposal, however, is the finding that this effect is often absent. A possible explanation for these absences is that familiarity may vary both within and across sequence types; therefore, not all consecutive sequences are necessarily more familiar than all nonconsecutive sequences. Accordingly, under this familiarity perspective, familiar sequences should always be processed faster than unfamiliar sequences, but consecutive sequences may not always be processed faster than nonconsecutive sequences. To test this hypothesis in an adult population, we used a comparative judgment approach to measure familiarity at the individual sequence level. Using this measure, we found that although not all participants showed a reverse distance effect, all participants displayed a familiarity effect. Notably, this familiarity effect appeared stronger than the reverse distance effect at both the group and individual level; thus, suggesting the reverse distance effect may be better conceptualized as a specific instance of a more general familiarity effect.

Walking to a number: is there affective involvement in generating the SNARC effect in numerical cognition?

The effect known as the spatial-numerical association of response codes (SNARC) documents fast reaction to small numbers with a response at the left and to large numbers with a response at the right. The common explanation appeals to a hypothetical mental number line of a left-to-right orientation with the numerical magnitudes on the line activated in an automatic fashion. To explore the possibility of emotional involvement in processing, we employed prototypical affective behaviors for responses in lieu of the usual spatial-numerical ones (i.e., of pressing lateralized keys). In the present series of experiments, the participants walked toward a number or walked away from a number (in a physical approach-avoidance setup) or said "good" or "bad" in response to a number. We recorded strong SNARC effects with affective responding. For example, it took participants longer to say "good" than "bad" to small numbers, but it took them longer to say "bad" than "good" to larger numbers. Although each particular outcome can still be accounted for by a spatial interpretation, the cumulative results are suggestive of the possibly of affective involvement in generating the effect.

Electrophysiological correlates of symbolic numerical order processing

Determining if a sequence of numbers is ordered or not is one of the fundamental aspects of numerical processing linked to concurrent and future arithmetic skills. While some studies have explored the neural underpinnings of order processing using functional magnetic resonance imaging, our understanding of electrophysiological correlates is comparatively limited. To address this gap, we used a three-item symbolic numerical order verification task (with Arabic numerals from 1 to 9) to study event-related potentials (ERPs) in 73 adult participants in an exploratory approach. We presented three-item sequences and manipulated their order (ordered vs. unordered) as well as their inter-item numerical distance (one vs. two). Participants had to determine if a presented sequence was ordered or not. They also completed a speeded arithmetic fluency test, which measured their arithmetic skills. Our results revealed a significant mean amplitude difference in the grand average ERP waveform between ordered and unordered sequences in a time window of 500-750 ms at left anterior-frontal, left parietal, and central electrodes. We also identified distance-related amplitude differences for both ordered and unordered sequences. While unordered sequences showed an effect in the time window of 500-750 ms at electrode clusters around anterior-frontal and right-frontal regions, ordered sequences differed in an earlier time window (190-275 ms) in frontal and right parieto-occipital regions. Only the mean amplitude difference between ordered and unordered sequences showed an association with arithmetic fluency at the left anterior-frontal electrode. While the earlier time window for ordered sequences is consistent with a more automated and efficient processing of ordered sequential items, distance-related differences in unordered sequences occur later in time.

Severe Developmental Dyscalculia Is Characterized by Core Deficits in Both Symbolic and Nonsymbolic Number Sense

A long-standing debate concerns whether developmental dyscalculia is characterized by core deficits in processing nonsymbolic or symbolic numerical information as well as the role of domain-general difficulties. Heterogeneity in recruitment and diagnostic criteria make it difficult to disentangle this issue. Here, we selected children (n = 58) with severely compromised mathematical skills (2 SD below average) but average domain-general skills from a large sample referred for clinical assessment of learning disabilities. From the same sample, we selected a control group of children (n = 42) matched for IQ, age, and visuospatial memory but with average mathematical skills. Children with dyscalculia showed deficits in both symbolic and nonsymbolic number sense assessed with simple computerized tasks. Performance in the digit-comparison task and the numerosity match-to-sample task reliably separated children with developmental dyscalculia from controls in cross-validated logistic regression (area under the curve = .84). These results support a number-sense-deficit theory and highlight basic numerical abilities that could be targeted for early identification of at-risk children as well as for intervention.

Extending ideas of numerical order beyond the count-list from kindergarten to first grade

Ordinal processing plays a fundamental role in both the representation and manipulation of symbolic numbers. As such, it is important to understand how children come to develop a sense of ordinality in the first place. The current study examines the role of the count-list in the development of ordinal knowledge through the investigation of two research questions: (1) Do K-1 children struggle to extend the notion of numerical order beyond the count-list, and if so (2) does this extension develop incrementally or manifest as a qualitative re-organization of how children recognize the ordinality of numerical sequences. Overall, we observed that although young children reliably identified adjacent ordered sequences (i.e., those that match the count-list; '2-3-4') as being in the correct ascending order, they performed significantly below chance on non-adjacent ordered trials (i.e., those that do not match the count-list but are in the correct order; '2-4-6') from the beginning of kindergarten to the end of first grade. Further, both qualitative and quantitative analyses supported the conclusion that the ability to extend notions of ordinality beyond the count-list emerged as a conceptual shift in ordinal understanding rather than through incremental improvements. These findings are the first to suggest that the ability to extend notions of ordinality beyond the count-list to include non-adjacent numbers is non-trivial and reflects a significant developmental hurdle that most children must overcome in order to develop a mature sense of ordinality.

When A is greater than B: Interactions between magnitude and serial order

Processing ordinal information is an important aspect of cognitive ability, yet the nature of such ordinal representations remains largely unclear. Previously, it has been suggested that ordinal position is coded as magnitude, but this claim has not yet received direct empirical support. This study examined the nature of ordinal representations using a Stroop-like letter order judgment task. If ordinal position is coded as magnitude, then letter ordering and font size should interact. Experiments 1 and 2 identified a significant interaction between letter size and ordering. Specifically, a facilitation effect was observed for alphabetically ordered sequences with decreasing font size (e.g., B C D). This suggests an overlap in the mechanisms for order and magnitude processing. The finding also suggests that earlier ranks may be represented as "more" in such a magnitude-based code, and vice versa for later ranks.

Common and distinct predictors of non-symbolic and symbolic ordinal number processing across the early primary school years

What are the cognitive mechanisms supporting non-symbolic and symbolic order processing? Preliminary evidence suggests that non-symbolic and symbolic order processing are partly distinct constructs. The precise mechanisms supporting these skills, however, are still unclear. Moreover, predictive patterns may undergo dynamic developmental changes during the first years of formal schooling. This study investigates the contribution of theoretically relevant constructs (non-symbolic and symbolic magnitude comparison, counting and storage and manipulation components of verbal and visuo-spatial working memory) to performance and developmental change in non-symbolic and symbolic numerical order processing. We followed 157 children longitudinally from Grade 1 to 3. In the order judgement tasks, children decided whether or not triplets of dots or digits were arranged in numerically ascending order. Non-symbolic magnitude comparison and visuo-spatial manipulation were significant predictors of initial performance in both non-symbolic and symbolic ordering. In line with our expectations, counting skills contributed additional variance to the prediction of symbolic, but not of non-symbolic ordering. Developmental change in ordering performance from Grade 1 to 2 was predicted by symbolic comparison skills and visuo-spatial manipulation. None of the predictors explained variance in developmental change from Grade 2 to 3. Taken together, the present results provide robust evidence for a general involvement of pair-wise magnitude comparison and visuo-spatial manipulation in numerical ordering, irrespective of the number format. Importantly, counting-based mechanisms appear to be a unique predictor of symbolic ordering. We thus conclude that there is only a partial overlap of the cognitive mechanisms underlying non-symbolic and symbolic order processing.

Numeral order and the operationalization of the numerical system

Recent years have witnessed an increase in research on how numeral ordering skills relate to children's and adults' mathematics achievement both cross-sectionally and longitudinally. Nonetheless, it remains unknown which core competency numeral ordering tasks measure, which cognitive mechanisms underlie performance on these tasks, and why numeral ordering skills relate to arithmetic and math achievement. In the current study, we focused on the processes underlying decision-making in the numeral order judgement task with triplets to investigate these questions. A drift-diffusion model for two-choice decisions was fit to data from 97 undergraduates. Findings aligned with the hypothesis that numeral ordering skills reflected the operationalization of the numerical system, where small numbers provide more evidence of an ordered response than large numbers. Furthermore, the pattern of findings suggested that arithmetic achievement was associated with the accuracy of the ordinal representations of numbers.

The Neural Representation of Ordinal Information: Domain-Specific or Domain-General?

Ordinal processing allows for the representation of the sequential relations between stimuli and is a fundamental aspect of different cognitive domains such as verbal working memory (WM), language and numerical cognition. Several studies suggest common ordinal coding mechanisms across these different domains but direct between-domain comparisons of ordinal coding are rare and have led to contradictory evidence. This fMRI study examined the commonality of ordinal representations across the WM, the number, and the letter domains by using a multivoxel pattern analysis approach and by focusing on triplet stimuli associated with robust ordinal distance effects. Neural patterns in fronto-parietal cortices distinguished ordinal distance in all domains. Critically, between-task predictions of ordinal distance in fronto-parietal cortices were robust between serial order WM, alphabetical order judgment but not when involving the numerical order judgment tasks. Moreover, frontal ROIs further supported between-task prediction of distance for the luminance judgment control task, the serial order WM, and the alphabetical tasks. These results suggest that common neural substrates characterize processing of ordinal information in WM and alphabetical but not numerical domains. This commonality, particularly in frontal cortices, may however reflect attentional control processes involved in judging ordinal distances rather than the intervention of domain-general ordinal codes.

Ordinality: The importance of its trial list composition and examining its relation with adults' arithmetic and mathematical reasoning

Understanding whether a sequence is presented in an order or not (i.e., ordinality) is a robust predictor of adults’ arithmetic performance, but the mechanisms underlying this skill and its relationship with mathematics remain unclear. In this study, we examined (a) the cognitive strategies involved in ordinality inferred from behavioural effects observed in different types of sequences and (b) whether ordinality is also related to mathematical reasoning besides arithmetic. In Experiment 1, participants performed an arithmetic, a mathematical reasoning test, and an order task, which had balanced trials on the basis of order, direction, regularity, and distance. We observed standard distance effects (DEs) for ordered and non-ordered sequences, which suggest reliance on magnitude comparison strategies. This contradicts past studies that reported reversed distance effects (RDEs) for some types of sequences, which suggest reliance on retrieval strategies. Also, we found that ordinality predicted arithmetic but not mathematical reasoning when controlling for fluid intelligence. In Experiment 2, we investigated whether the aforementioned absence of RDEs was because of our trial list composition. Participants performed two order tasks: in both tasks, no RDE was found demonstrating the fragility of the RDE. In addition, results showed that the strategies used when processing ordinality were modulated by the trial list composition and presentation order of the tasks. Altogether, these findings reveal that ordinality is strongly related to arithmetic and that the strategies used when processing ordinality are highly dependent on the context in which the task is presented.

Symbolic Number Ordering and its Underlying Strategies Examined Through Self-Reports

Symbolic number ordering has been related to arithmetic fluency; however, the nature of this relation remains unclear. Here we investigate whether the implementation of strategies can explain the relation between number ordering and arithmetic fluency. In the first study, participants (N = 16) performed a symbolic number ordering task (i.e., “is a triplet of digits presented in order or not?”) and verbally reported the strategy they used after each trial. The analysis of the verbal responses led to the identification of three main strategies: memory retrieval, triplet decomposition, and arithmetic operation. All the remaining strategies were grouped in the fourth category “other”. In the second study, participants were presented with a description of the four strategies. Afterwards, they (N = 61) judged the order of triplets of digits as fast and as accurately as possible and, after each trial, they indicated the implemented strategy by selecting one of the four pre-determined strategies. Participants also completed a standardized test to assess their arithmetic fluency. Memory retrieval strategy was used more often for ordered trials than for non-ordered trials and more for consecutive than non-consecutive triplets. Reaction times on trials solved by memory retrieval were related to the participants’ arithmetic fluency score. For the first time, we provide evidence that the relation between symbolic number ordering and arithmetic fluency is related to faster execution of memory retrieval strategies.

Children's Knowledge of Symbolic Number in Grades 1 and 2: Integration of Associations

How do children develop associations among number symbols? For Grade 1 children (n = 66, M = 78 months), sequence knowledge (i.e., identify missing numbers) and number comparison (i.e., choose larger number) predicted addition, both concurrently and indirectly at the end of Grade 1. Number ordering (i.e., touch numbers in order) did not predict addition but was predicted by number comparison, suggesting that magnitude associations underlie ordering performance. In contrast, for Grade 2 children (n = 80, M = 90 months), number ordering predicted addition concurrently and at the end of Grade 2; number ordering was predicted by number comparison, sequencing, and inhibitory processing. Development of symbolic number competence involves the hierarchical integration of sequence, magnitude, order, and arithmetic associations.